Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

44 marcus giaquintoof knowing structures; in some cases we can know structures in more intimateways, by means of our visual capacities.We can say, broadly, that our ability to discern the structure of simplestructured sets is a power of abstraction; but we do not at present have anadequate account of the operative faculties or processes. That we have anability to discern structure is made evident by our manifest ability to spot astructural analogy. A striking case is provided by the history of biology. Thepattern of gene distribution in dihybrid reproduction postulated by Mendeliantheory can be found in the behaviour of chromosomes in meiosis (division of cellsinto four daughter cells, each with half the number of chromosomes of theparent) and subsequent fusion of pairs of daughter cells from different parents.The hypothesis that chromosomes are agents of inherited characteristics issuedfrom this observation, and was later confirmed.³Somehow biologists spotted the common pattern. But how? For an answerone might look to the large cognitive science literature on analogy. Thissubject is still at an early stage, with lots of speculation and many theories.Among the most promising are those based on the structure-mapping idea ofDedre Gentner.⁴ This, however, is no good for present purposes, because ittakes cognition of structure as given, and cognition of structure is preciselywhat we want to understand.I will proceed by suggesting some possibilities consistent with what I knowof current cognitive science, in particular the science of vision and visualimagination. I will restrict consideration to just a few structures, mostly verysimple finite structures. But I will also say something about our grasp of infinitestructures.2.2 Visual grasp of simple finite structuresHere is a simpler biological example of a structured set. Consider the setconsisting of a certain cell and two generations of cells formed from that initialcell by mitosis, division of cells into two daughter cells, under the relation ‘xis a parent of y’. In this structured set there is a unique initial cell (initial, inthat no cell in the set is a parent of it), a first generation of two daughters and asecond generation of four grand-daughters, which are terminal (as they are notparents of any cell in the set). How can we have knowledge of the structure ofthis structured set?³ Priority is attributed to W. Sutton (1902). Others were hard on his heels. ⁴ Gentner (1983).